I A--16! P4 - 11729inis.jinr.ru/sl/NTBLIB/JINR-P4-11729.pdf · A6paMoB Jl.l1. H np. P4 - 11729...
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Transcript of I A--16! P4 - 11729inis.jinr.ru/sl/NTBLIB/JINR-P4-11729.pdf · A6paMoB Jl.l1. H np. P4 - 11729...
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P4 - 11729
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Hanpae/leHO e "Journal of Physics B"
A6paMoB Jl.l1. H np. P4 - 11729
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B 38llB'I"e JlBYX UeHTpOB KBBHTOBOA M9X8HBX.B Bbi'iHCneHbl cfla3bi pacceSIHH
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Abramov D.I. et aJ. P4-11729
Phase Shifts in the Coulomb Two-Center Problem
The phase shifts in the Coulomb two-center problem of quantum mechanics are calculated and the asymptotic expansions for them are found in the limit of large and small distances between the charges. The asymptotic expansions for the phase shifts are compared with the results of direct numerical calculations. The relations between the one- and two-center phase shifts are considered.
The investigation has been performed at the Laboratory of Theoretical Physics, JINR.
Preprint of the Joint Institute for Nuclear Research. Dubna 1978
© 1978 06'bei!BBeHHbliJ BHCTHTyT H/!epHbiX BCCllei!OBHBBil /1y6HH
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0,2 -2,367306 0,144738 -3,618769 -0,145791 -2,366636 0,145478 -3,618682 -0,14362
0,3 -2,419958 0,088451 -3,598133 -O,II7386 -2,412442 0,096732 -3,59764 -0,1065
0,4 -2,479047 0,026586 -3,569970 -0,087965 -2,4394 0,0700 -3,5682 -0,054
0,5 -2,541370 -0,036692 -3,535229 -0,064729 -2,404 O,II36 -3,5303 O,OI
a) l{yJIOHOBCKM was a e>,_+.,. ( k) B) ~asa, BU~CJieHHM COrJiaCHO aJirOpHTMY paOOTU/2/
c) ~asa, BU~CJieHHM IIO aCHMIITOTH'leCIOO'II tflopMyJiaM (I4) H (I6).
ACHUD'l'OTHXS p.K.C .If!. fl.,., ( c, et, }. ; f) HII88T BH,l!
"'.z n 1004. ( c, Q, :\;f) : (=iz~ JI'O\ (c .i!) ~I= 'dr, r ' 'il J'
(IS)
rAe J,.,. (c>~) - ~YHKUHft Eecc6~ft, a apryM6H'1' ~YHKUHH
13
TadJinua 2. CDasu pacceHIIIDI /),. 01 (k,R) rrpn R- 0 ,IJ.7IJI CJIY'<M Z~=1., Zt Z.
R"l] --~2---T- -~~ --l- - I l 5
0 -27,904823a) -9,824298 -2,302397 -0,267676
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-9,8!5888 -9,823567
0,2 -27,872433 -9, 79!048 -27,905048 -9,82!376
0,3 -27,832940 -9,75!438 -27,905330 -9,8!7725
0,4 -27,78!97! -9,700424 -27,905725 -9,8!26!2
-2,293780 -2,3!09!5
-2,268552 -2,336472
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-0,2552!2 -0,244266
-0.22!393 -0,!74087
-0,!76348 -0,066988
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0,5 -27,722986 -27,906232
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HHH
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I.
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+ c-1" T Qft.(. tJ ~ r L-:1. &
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14
1 I
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Tac5JIMua 3. <liasu pacceHHMH D.,..'J- (k, R) rrpn k-+oo, R-oo ,IJ.7IJI
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5,0
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-6,5!67 !,3649
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a) rJ!a3a, Bbl'1HCJI8HHM corJiacno aJiropnTr.zy- pa60Tul21 . B) wasa , Bhl'll!CJieHHaJI no acrnmTOTH'iecKoii <floptJY.'Ie (20).
15
llpe.zten:&HH~ nepexo~ I - oo B ilJOpMyne (19) C Y'IBTOM SCHKn-
TOTH'IBCKoro noBe~eHHR ilJYHRUH~ EeccenR npHBO~HT K cne~D~euy
BNPSKBHHD AnR WSSN A ""'f ( lc, ~)
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(20)
= - z~ + Zz. e.,. k R + L { zj -t-Zz. (2.'-+J.)- z:;r z:t 'Zz l_ IV:&'!
I< t, lcR I< r kl. J ~v.s- J
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16
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(+) TeHUHane - Zjr, Z. = Z :l ( CK. ( 7)), a BenH'IHHa ~ ..;.,_
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(23)
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(25)
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17
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B npHBe~eaa~ npHYepax pacxo~~eaHe aaOnv~aerca B 5-7 aaa&e,
qro CBH~eTeRbCTByeT 0 B~COKO~ TOqHOCTH anropHTY8, peanH30B8HHO
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6. CBH3b MEllY PA3AMH ~YXUEHTPOBOR H O~HOUEHTPOBOR RYJIOHOBCimX 3A,nA"l!
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npH HeKoTop~x aaaqeaHax R p.K.c.~. H y.K.c.w. coBna~aiDT c
O~HOU6HTpOBHYH KYHOHOBCKHYH WYHKUHHYH B cwepoH~8RbHNX KOOp~HH8-
T8X H YOryT 6NTb npe~CTaBneHN nOnHHDMaMH. QqeBH~HO, qTo B Henpe
pNBHOY cneKTpe y.K.c.w. BCer~a COBna~aeT C O~HOUeHTpOBD~ yrno-
BD~ WYHKUHeA ~na aapa~a ~ = ~ - ~ • Kpoue Toro, KaK 2. j
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aaaqeaHA k H R, npH KoTopHx p.K.c.w. coBna~aiDT c o~aoneaTpOBHMH KYROHOBCKHYH ~YHKUHHYH B C~epoH~anbH~X KOOp~HH8T8X ~Jlli sapH~a z :: z .1 + z~ . napa T8KHX SHaqeaH~ ( k.,, R.,) onpe~eHH6TCH H3 YCROBHH
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19
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k•0.2
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1
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BOft c zj = zz. = .1) H o,lUiou.ea'l'poBot (z1 =o, z2 =~) 38~'11 npH k = 0,2; 0,6; 1,0; 2,0.
20
T. 6. B TOIIK8X ( ko, ~o) ~YX!leHTpOB8fl \IJ838 fl ,..,.. ( lc, R) MolleT CiiiT:& BliiiHCneaa no !}:op.tyne (25) ~Jifl 0,11-
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( k, R)
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C k, R.")
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KDC'l'H ( k, R) • lliiHHR U6peC6lJ6HH.fl STHX .nBYX ll0B6PXHOCT6ft
30
m=o. q·1 Z 1=0 Z2 =2
L: P.ac. I3. IloBepxaocu - >.:J ( k, R) coCic'l'B&aHI\IX 3Ha'iead
O,ttliOnaH'l'poBOft 38.11,8"1H ( Z.:l = 0, Z~ = Z) B c<fj&poH,l1,8.JI:&
HI\IX R00pli,HH8'1'8X H8 UJIOCKOC'l'H ( k, R).
21
k:
m =o, q =1
z, =Z2 =1
Puc. !4. TioBepxaocn - ~ 01 ( k, R) co6c'rB6HHHX saa'lead
mxneattpOBOft 38,n8'1111 ( z .1 = Z 2. = 1) R8 IlJIOCliOCTH
( k > R) • Ha pHCYHKe YR838H8 'r8RJt6 JmR.I'l.fl ( ko, Ro) nepece'leaKR noBepxaoctteft - >-o:1 ( k, R) x ).:1 ( k, ~).
onpe,li,6Jl.R6T RPHBYD ( ko, R. 0 ) , H8 RO'rOpoi <f83H .6..,.} ( k, R)
mxueR'rpOBOI 38,n8'1lll MOryT 6HT:& BH'IHCJ18HH B BH,ll,6 PJJ,li,OB (22),
(25), npe,ll,cTaBJLRD!UiX <fasy L\ 0
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sa,natJ•.
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2. ~ j >2. >2
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a cyMMDft O,li,HOU6HTpOBHX <fas 8 "'} c S8p!l,li,8MH Z.:~. =-0, z:z.:: :f a Z.s. = 0, Z.z. =3 coottBeTCttBeRRo.
22
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2,0
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~ , ' ey.nOHOBCKliX WS3
k I z~ = 1 I Z 1 =o I Z 1 =0 I Z 1 =1 I £ z .... 3 Z.; = (! z .. =3 z, = 0
0,2 -5,1892 -27,9053 -42,2169 -42,2048 9,II03 0,4 -I, 7327 -9,8237 -15,2733 -15,2614 3,7050 I,O -0,4833 -2,3009 - 3,6336 - 3,6225 0,8383 2,0 -O,?I86 -0,8169 - 1,2329 - 1,2230 0,1875 5,0 -0,0851 -0,2603 - 0,3608 - 0,3528 0,0074
IO,O -0,0376 -O,II45 - 0,!547 - 0,1480 -0,0041 20,0 -O,OI3I -0,0396 - 0,0527 - 0,0489 -0,0038
0,2 -5,I869 -27,8999 -42,2104 -41,9965 8,9097 0,4 -1,7280 -9.8I2I -15,2597 -15,0509 3,5108 1,0 -0,4717 -2,2721 - 3,6000 - 3,4099 0,6660 2,0 -0,1964 -0,7613 - I,I679 - I,OII6 0,0539 5,0 -0,0415 -0,1537 - 0, 2342 - 0,1740 -0,0212
IO,O -0,0176 -0,0607 - 0,0894 - O,II60 0,0377 20,0 -0,0579 -0,!736 - 0,23!5 - 0,2410 0,0095
0,2 -5,1802 -27,888! -42,2028 -41,9409 8,8726 0,4 -I, 7!49 - 9,7927 -15,2441 -14,9930 3,4854 I,O -0,4405 - 2,2255 - 3,5627 - 3,3555 0,6894 2,0 -0,1452 - 0,6829 - 1,1060 - 0,989! O,I6IO 5,0 -0,0372 - 0,1664 - 0,2714 - 0,3578 0,1542
!0,0 -O,II58 - 0,3471 - 0,4628 - 0,4989 0,0360 20,0 -0,1064 - 0,3192 - 0,4256 - 0,4287 0,0031
0,2 -5,1608 -27,8836 -42,2II8 -42,3824 9,3380 0,4 -1,6769 - 9,7859 -!5,26!3 -!5,4283 3,9654 I,O -0,4405 - 2,2I7I - 3,6094 - 3,7786 I,I209 2,0 -0,09!2 - 0,7250 - 1,2286 - !,4369 0,6208 5,0 -0,23!5 - 0,6941 - 0,9259 - 1,0564 0,1308
IO,O -0,2128 - 0,6385 - 0,8513 - 0,8633 0,0!20 20,0 -0,!458 - 0,4373 - 0,5834 - 0,5844 0,0013
23
Taogrrga 5. (ITpo~o~eH:ae) Tao.~nma 6 ( rrpo.n;oJDKeHRe)
I i!" = 0 I I i!" = :1 I I I I I k I z" .. t Z.1 = o R I I Z"::1. I Z 1 =0 'Z" = 0 'Z"=1 E. R E. k Za,=O z., .. ~ :z.,.{t Z1 •3 z,=o z.~ = 3 .z, ... ~ z, .. .3
5,0 0,2 -5,I274 -27,9783 -42,3324 -43,5075 IU,40I8 3.0 -0.5673 -I.7067 -2.28II -2.2790 0.0050 0,4 -I,62I4 - 9,9705 -I5,504I -I6,5705 4,9786 5.0 -0.3386 -I,OI67 -I.3568 -I,3548 -0.0005 I,O -0,4034 - 2,6653 - 4,I893 - 4, 977I I,9085 IO.O -O.I650 -0.495I -0.6604 -0.6582 -O.OOI9 2,0 -0,5785 - I, 7582 - 2,3994 - 2,9690 0,6324 20.0 -0.07I6 -0.2I30 -0.2846 -0.2832 -O.OOI4 5,0 -0.4782 - I,4346 - I,9I28 - I,9482 0,0354 I.O 0.2 -9.09IO -33.2755 -47.8069 -47. 793{) 5.4265
IO,O -0,3I55 - 0,9467 - I,2622 - I,2663 0,004I 0.4 -4.345I -I4.3859 -20.22I3 -20.2089 I.4779 20,0 -O,I939 - 0,5822 - 0,776I - 0,7768 0,0007 r.o -I. 7094 -5.248I -7.I2I5 -7 .IIII O.I537
2.0 -0.8494 -2.5643 -3.437I -3.4279 O.OI42 IO,O 0,2 -5, 2200 -28,2477 -42,6334 -44,7429 II,2752 3.0 -0.5624 -I .69I9 -2.26I4 -2.2526 -O.OOI7
0,4 -I ,8I68 -I0,4974 -I6,0962 -I7,92I6 5,6075 5.0 -0.3300 -0.99I2 -I.3227 -I.3I37 -0.0075 I,o -I,I59I - 3,8556 - 5,5224 - 6,7355 I,7208 IO.O -O.I433 -0.4298 -0.5730 -0.559I -O.OI40 2,0 -I,05I8 - 3,I899 - 4,2504 - 4,52IO 0,2793 20.0 -0.0532 -O.I602 -0.2I34 -0.2255 -o.orao 5,0 -0,63II - I,8933 - 2,5244 - 2,5407 O,OI63
2.0 0.2 -9.0897 -33.27I2 - -47.8079 -47.7529 5.3920-IO,O -0,3882 - I,I645 - I,5524 - I,5544 O,OOI7 0.4 -4.3425 -I4.378I -20.2I04 -20.I600 I.4394 r.o -I. 7029 -5.2286 -7.0954 -7.0526 O.I2II 2.0 -0.836I -2.5241 -3.3832 -3.3436 -0.0!66 Ta6.~nma 6 ' 3.0 -0.54I3 -1.6283 -2.I76I -2.I344 -0.0353
CpaBHeHHe ,!Ulyxi.\8HTp0BHX Aos ( lc,R) :a O.n;HOD;eHTpoBHX 5.0 -0.2866 -0,8594 -I.I456 -I.0836 -0.0623 ro.o ...Q,I066 -0.3234 -0.4352 -0.4761 0.0460 AooS'( k,lt) KY~OHOBOKKX ~3 paoceHHHH 20.0 -O.I3I9 -0.3956 -0.5275 -0.5296 0.002!
5.0 0,2 -9.0807 -33.2427 -47.7637 -47.3657 5.0423
R I k I Z, =I I Zs :::0 I z~ :::O I Zs =I
I 0.4 -4.3243 -I4.32I9 -20.I33I -I9. 7573 I.IIIO
z~,:::O l& =4 z..~=3 f r.o -!.6553 -5.0798 -6.8893 -6.5I67 -0.2I83 z,. =3
2.0 -0.7I62 -2,I5I3 -2.8868 -2.4267 -0.4408 O,I 0,2 -9,09I5 -33,2777 -47,8!09 -47.8!08 5.44I6 3.0 -0.3053 -I.04I9 -I.493I -I.3700 0.0228
0,4 -4,3460 -I4,3885 -20,2243 -20.2242 I.4897 5.0 -0.35I6 -1.0520 -!.3998 -I.4998 0.09~ I,O -I, 7II5 -5,2545 -7 ,I300 -7.I299 O.I639 ro.o -0.2950 -0.885I -I .I80I -I.I858 0.0056 2,0 -0,8537 -2,577I -3,4542 -3,454I 0.0233 20.0 -O.I895 -0.5686 -0.7580 -0.7588 0.0008 3,0 -0,5688 -I, 7II3 -2,2873 -2.2872 0.0070 IO.O 0.2 -9.0460 -33.I274 -47.6II8 -45.8I45 3.6410 5,0 -0,34II -I,0245 -I,3672 -I.36?2 O.OOI6 0.4 -4.2566 -I4.0850 -I9.8332 -I8.2684 -0.0732
IO,O -O,I704 -0,5II3 -0,68I9 -0.68I8 O.OOOI I.O -1.4320 -4.4773 -6.2793 -5.6292 -0.280I 20,0 -0,0849 -0,2546 -0,3395 -0.3395 0.0000 2.0 -0.550I -I.9388 -2.7897 -3.I336 0.6447
0,5 0,2 -9,09I3 -33,2766 -47,8IOO -47.8066 5.4386 3.0 -0.7203 -2.I572 -2.87I6 -3.0274 O.I499 0,4 -4,3457 -I4,3879 -20,2234 -20.2203 I.4867 5.0 -0.590I -1.7702 -2.3602 -2.3825 0.0222 I,O -I, 7IIO -5,2529 -7,I279 -7.I254 O.I6I5 ro.o -0.379I -I.I374 -I.5I66 -I.5I89 0.0024 2,0 -0,8527 -2,5740 -3,4501 -3.4448 O.OI8I
24 25
H3 e~ax ~a6nHn BH~o. ~~o:
a) npH k _. 0 np.a BC8X SH8~8HIUIX R. .DJ3YXnea~poBBSI <tasa
l:l.rn,_ 6.nue R coo~:seroTBYICIIle! el o~onea~po:eol ~ase c cyM-
Mapam~ sap.R.lJP.M Z = Z 1 + Z 2 , ~eM R cyMMe o~onea~po:sux
~88 C 38Pfl,nall.l! Z : 2 1 H Z = Z R. • 3~0 H ec~eC~BSHBO,
T8R R8R IIPH k - 0 ,IVUIB.8 BOJIH!l CYUI8C'!'B8HHO 6o;u,me /(
X8p8RT8pHOrO pasMepa C.I!CT8Mll,H B8RH~HH8 ~8Sll HeqyBCTBRT6RLHa
R CTpyR'l'ype CHCT8M~ I;
6) IIPH k -- oo t ..... 0 ~.a. mxneaTpo:san ~a sa~- (z1
=1. z =3" ' . }' ,_ z. ') C'1'8HOB.Iti'l'C.Il 6RHlt8 R CyMMe b.:". (Z1"' 0 Z = 1.) + 6,_0 (Z =O Z = 3)
0 y ' 2. ... , 1 ' z 'iSM R ~ ... ,_(Z.1=0, Z2. == ~). 3'1'0'1' ~8R'1' T8Rit8 8C'l'8C'!'B8H8H, •
DOCJ!tOJU,Icy IIpH k _. oo p8CC8.RHH8 H8 COC~8BHOI CHC~8M8 MOZB.O
p8CCM8'!'pHB8'l'L R8l!t IIOCH6~0B8T8HLH08 H838BHCHM08 p8CC8.RHH8 H8
R8lt,liPM HS neH'l'pOB CHC~eMaf8 • 91 B) npH :scex k 11 .R ,D;ByxneaTpo:san ~838 A""} 6.llHSRa R
coo~:snCT:BYICIIlel el o~onea~po:sol lf:ase !I rn1- c cyMM8pH'hllll
saPfi,liPM z = z.1 + zt. r) pasaocn Ll
0
t"".(Z =0 z :4)-[ll" (z =oz =3"+!{ (:z:0"->1 y 1 ' 2 "''j.. J. ' 2 ') ,..,_ ~ ,~.£ .. 1
c.na6o saBHCH~ o~ R H onpe,neJIHSTC.R :s ocso:eaou napaueTpoM k,
IIp.!!'q8N npH k -.oo YR8S8B.H8.R p8SROC'!'L CTpeMJl'l'C.Il R fiYJlll.
Ha p.ac. 15 rrp.aBe.n;eHH rpaqmKH .n;ByxneaTpoBHX <Pas Ll,.'l- ( k, R.) ,IJ,1Ui ClWTeMll Z.1 = Zz = 1. KaK qiyH~ R. IIPH <j)HKCHPOBaHHHOM
saa~emm k = 1 H pas.mP:rnHX aa6opax KBaHTOBHX 'Ul:ce.n n~ H 1- . Ha 9TOM pHCyaKe IIpOCJieXHBaroTC.R OCHOBH!le OC06eHHOCTH IIOBe.n;eHHH
<Pas paccemm.s:r. Hs rpaqmKa .nerKo TaKJKe mt~eT:&, "'ITO rrpH ~ .6 '1-<Pasll f1 "'} ( lc, /?.) , XOTR 6bl B O.n;HOH TO "'iKe IIpH KOH8"'iHHX ({
COBIIa,IJ;aroT C 1\YJIOHOBCKHMI!: <P83aMH f>e{k) ,IJ,1Ui saprr.n;a Z = Z.1 ~ Z.lt.,
T. e. CO CBOHMJ 3Ha"'ieHH8M .L\ ,...,. ( lc, 0) IIpH R. = 0
26
i llm'l Ck.Ri R
0 5 10 15 20
z1 =Z2·1 k =1 t =q•m -1
-2~'\
-3
-4-1!•7 -
-s
he. I5. )J;syxneHTpoBHe <Pasll A "'t- ( lc, R) ,IJ,7Ui CHCTeMll
Z = Z :::- :J. ITpP! 3H8"'i8Hlm k,. :f ; f.:: rn +4 • :1 ll. y
B saB~eaae SBTOPll npBBOCR'l' r.ny6onym 6~aro~paoc~~
H.B.Kouapo:sy sa QaRHHe o6cya,neaas.
JlHTEPATYPA
I. H.B.KoMapo:s, .n.H.IIoaor.aapeB, C.D.CJJSB.RROB. C¢epo.w.na.n:&BHe 11
xy.noHOBCBHe c~epoH.na.n:&HHe ~Imnn. M., Hayxa, 1976. 2. L.I.Ponomarev and L.N.Somov, J. Camp. Phys., 20,
183, (1976).
3. D.I.Abramov, S.Yu.S1avycnov, J. Phys. ~. 238 (1978).
4. ~.H.A6paMOB, H.B.KouapoB. TM~~. 235 (1976). 5. P.Th.Green1and, W.Greiner. Theoret. Chim. Acta (Ber1)
1 42, 27 (1976).
6. D.H.neMRoB. llHC:&Ka B XJTw, 1,IOI(I968).
27
?. D.H.~MROB, B.C.Py~ROB, 13T~. 59, 2035, 19?0. 8. R.J.G1auber, High-Energy co11ieion.theory, Lectures in
the oretica1 physics, Boulder, 1958 (Interscience, New York, 1959).
9. H.Fauchiotti and T.h.Oeborn, Nuc1. Phye. A177 (1971),
273-288.
PyKOUMCb ITOCTYITHRa B M3~aT8RbCKHa OT~en
4 MDRII 1978 ro~a.
28
